Growth and shape transition of small silicon self-interstitial clusters

Transcription

1 Growth and shape transition of small silicon self-interstitial clusters Sangheon Lee and Gyeong S. Hwang* Department of Chemical Engineering, University of Texas, Austin, Texas 78712, USA Received 12 January 2008; revised manuscript received 8 May 2008; published 14 July 2008 The growth behavior of small self-interstitial clusters in crystalline Si is presented based on extensive combined Metropolis Monte Carlo, tight-binding molecular dynamics, and density-functional theory calculations. New stable structures for small interstitial clusters I n,5 n 16 are determined, showing that the compact geometry appears favored when the cluster size is smaller than 10 atoms n 10. The fourfoldcoordinated dodecainterstitial I 12 structure with C 2h symmetry is identified to serve as an effective nucleation center for larger extended defects. This work provides the first theoretical support for earlier experiments which suggest a shape transition from compact to elongated structures around n=10. DOI: /PhysRevB PACS number s : J I. INTRODUCTION There have been significant efforts to understand the fundamental behavior of Si interstitial defects created by bombardment with energetic dopant ions, due to their crucial role in defining ultrashallow pn junctions for ever smaller semiconductor device fabrication. Single Si interstitials are highly mobile even at room temperature. 1,2 Hence, in bulk Si, selfinterstitials are likely to remain in the form of clusters or interstitial-impurity complexes. The formation and structure of rodlike 311 defects have been well characterized by high-resolution transmission electron microscopy HRTEM. 3 5 In addition, a series of recent spectroscopy measurements 6 11 have evidenced existence of small compact self-interstitial clusters before they evolve into larger extended defects. In ultrashallow junction formation with low-energy implanted dopants, such small interstitial clusters are thought to be a main source for free interstitials responsible for dopant transient enhanced diffusion and agglomeration during postimplantation thermal treatment Hence, significant experimental and theoretical efforts have been made to determine the structure and stability of small self-interstitial clusters as well as their growth to larger extended defects, yet still unclear. Earlier inverse model studies 42,43 based on experimental observations of the spatial and temporal evolution of extended defect distribution in ion-implanted Si suggested the occurrence of structural transition between small compact clusters and larger extended defects when the cluster size is around 10 atoms. This prediction has been supported by a series of low-temperature photoluminescence PL studies 7,9,10 which show the signatures of small compact clusters of various sizes in ion-implanted Si upon short annealing. In addition, the PL measurements demonstrate changes in the spectra from multiple sharp peaks to broad but distinct signatures during prolonged annealing, ascribed to the shape evolution of self-interstitial clusters from compact to extended forms. While the empirical studies are rather limited to explicitly show the formation and atomic structure of small compact interstitial clusters, a few previous theoretical studies 30,33,40 also predicted the possibility that a chainlike, elongated tri-interstitial cluster may play a role for the compact-to-extended transition. Only a few small compact interstitial clusters I n,n=2 4 have been unequivocally identified by first-principles calculations. 27,28,33 At present, still the atomic structure and stability of larger compact clusters n 5 are uncertain due largely to the possible complexity in their geometries. The lack of information hampers revealing how small compact interstitial clusters grow to larger extended defects. In this paper, we present the growth and structural evolution of Si self-interstitial clusters in Si based on a combination of continuous random network model based Metropolis Monte Carlo CRN-MMC, tight-binding molecular dynamics TBMD and density-functional theory DFT calculations. We also discuss the growth of extended 311 defects from the small clusters. II. CALCULATION METHODS All atomic structures and energies reported herein were calculated using a plane-wave basis set pseudopotential method within the generalized gradient approximation of Perdew and Wang GGA-PW91 Ref. 44 to densityfunctional theory, as implemented in the well-established VI- ENNA AB INITIO SIMULATION PACKAGE VASP. 45 Vanderbilttype ultrasoft pseudopotentials 46 were used for core-electron interactions. Outer electron wave functions are expanded using a plane-wave basis set with a kinetic-energy cutoff of 160 ev. The Brillouin-zone sampling was performed using the Monkhorst-Pack mesh. We used the supercell approach for the defect calculations, with a fixed Si lattice constant of Å as obtained from volume optimization. Special care was taken to ensure that each supercell size is large enough to accommodate a given interstitial cluster with no significant interaction with its periodic images. For each defect system, all atoms were fully relaxed using the conjugate gradient method until residual forces on constituent atoms become smaller than ev/å. For TBMD simulations, semiempirical potentials developed by Lenosky et al. 47 were used. A Keating KT -like valence bond model 48 was employed for CRN-MMC calculations. Within the Keating-like valence force model, the strain energy E strain is given as /2008/78 4 / The American Physical Society

2 PHYSICAL REVIEW B 78, 共2008兲 SANGHEON LEE AND GYEONG S. HWANG III. RESULTS AND DISUCSSION A. Determination of small compact clusters FIG. 1. 共Color online兲 Ground state configurations for 共a兲 single-, 共b兲 di-, 共c兲 tri-, and 共d兲 tetrainterstitial defects in Si. Grey 共gold兲 balls indicate more distorted atoms than the rest of the lattice atoms 共in white兲. Corresponding defect symmetries are also indicated. Estrain = 1 1 kb共bi b0兲2 + 兺 k 共cos ij cos 0兲2, 兺 2 i 2 i,j where bi is the ith bond length and ij is the bond angle between bonds i and j, and the equilibrium and force constants are bo = Å, 0 = 109.5, kb = ev/ Å2 and k = ev. A detailed description of KT parameter optimization can be found elsewhere.41 For the sake of comparison, we first determined the lowest-energy configurations of the single-, di-, tri-, and tetrainterstitial clusters in the neutral state. Our calculations show the ground-state 具110典-split, C1h, C2, and D2d structures for I, I2, I3 and I4, respectively, as also predicted by previous theoretical studies.25,27,28,33 Here it is worth pointing out that our CRN-MMC calculations consistently predict the fourfold-coordinated C2 and D2d configurations for I3 and I4, respectively, as shown in Fig. 1, irregardless of the initial positions of interstitials so long as they are placed reasonably close to each other. This indicates the effectiveness of the CRN-MMC approach for determining fourfold defect structures in which all atoms have fourfold coordination. Figure 2 shows minimum-energy structures which we have identified for I5 I12 using a combination of CRNMMC, TBMD, and DFT-GGA calculations. For each cluster size, we first constructed possible fourfold-coordinated structures using CRN-MMC simulations, followed by TBMD simulations at high temperatures 共 1000 K兲 to check their thermal stability. Then, using DFT-GGA calculations we refined the geometries of the identified stable clusters, and compared their formation energies to determine the lowestenergy configuration among them. The combined approach has been proven to successfully determine minimum-energy configurations for Si self-interstitial clusters 共In, n ⱖ 3兲, particularly when they prefer fourfold coordination.41 Here, for FIG. 2. 共Color online兲 Predicted minimum-energy configurations for 共a兲 penta-, 共b兲 hexa-, 共c兲 hepta-, 共d兲 octa-, 共e兲 ennea-, 共f兲 deca-, 共g兲 hendeca-, 共h兲 dodecainterstitial defects in Si. For each defect, the left and right panels show two different views, as indicated. The symmetry of each defect is also indicated. Grey 共gold兲 balls represent more distorted atoms than the rest of the lattice atoms 共in white兲

3 GROWTH AND SHAPE TRANSITION OF SMALL SILICON each cluster size we only present the lowest-energy state among several local minima identified from our extensive search. Other stable structures can be found elsewhere. 41 Our results show that small interstitial clusters tend to favor compact structures, but the compact geometry is no longer favorable when the cluster size is greater than 10 interstitials. All atoms in the hendecainterstitial I 11 Fig. 2 g and dodecainterstitial I 12 Fig. 2 h clusters are fourfold coordinated, and they are elongated along the 110 direction. While the I 11 cluster exhibits a somewhat asymmetric shape, the I 12 structure is perfectly symmetric and less strained. The I 12 structure has a mirror symmetry with respect to the 110 plane perpendicular to the C 2 rotation axis along the 110 direction. Having four interstitials in each unit, the core part Fig. 5 a is characterized by four adjacent six-membered rings surrounded by five-, six-, and seven-membered rings. The 110 edge atoms are also fourfold coordinated, and their bond angles and lengths vary from 98.8 to and from to Å, respectively; insignificantly deviate from the equilibrium values of and Å in crystalline Si. The I 12 formation energy is predicted to be 1.60 ev per interstitial, which is substantially lower than 3.80 ev for the 110 -split single interstitial. We also find that the octainterstitial I 8 cluster comprising two stable compact I 4 clusters is very stable. For the lowest-energy I 8 structure, two intact I 4 structures are placed next to each other along the 110 direction, as shown in Fig. 2 d. The I 4 I 4 alignment with C 2v symmetry yields two eight-membered rings in the middle, which turns out to lower the induced strain, compared to two isolated I 4 clusters. While there is an energy variation with the I 4 I 4 alignment, 41 the predicted ground-state structure of I 8 is about 0.71 ev, more favorable than the case where two I 4 clusters are fully separated. Due to the high thermal stability of the I 4 and I 8 structures, the pentainterstitial I 5 Fig. 2 a and enneainterstitial I 9 clusters Fig. 2 e favor the I 4 +I and I 8 +I geometries, respectively, where the additional single interstitial is located near their host I 4 or I 8 cluster. For the hexainterstitial I 6 Fig. 2 b and heptainterstitial I 7 clusters Fig. 2 c, the combined I 4 +I 2 and I 4 +I 3 configurations appear energetically favored, with energy gains of 1.16 and 0.37 ev, respectively, over their fully separated counterparts, i.e., isolated I 2, I 3, and I 4 accordingly. Similarly, a stable I 8 +I 2 configuration is identified for the decainterstitial I 10 cluster, but turns out to be 0.36 ev less favorable than the stable fourfoldcoordinate structure as shown in Fig. 2 f. Our calculations also predict the I 8 +I 3 and I 8 +I 4 structures to be 0.93 and 1.59 ev, less stable than the elongated, fourfold-coordinated I 11 and I 12 structures, respectively. B. Compact-to-elongated transition Figure 3 summarizes the calculated formation energies of small interstitial clusters. Here the formation energy per interstitial E f n is given by: E f n = E n + N 1+n/N E N /n, where E n+n and E N are the total energies of N-atom supercells with a n-interstitial cluster and with no defect, FIG. 3. Calculated formation energies per interstitial E f of interstitial clusters shown in Figs. 1 and 2 indicated as This work as well as chainlike elongated structures indicated as Chainlike as a function of cluster size n. For the chainlike case, the atomic structures from Ref. 30 were recalculated within DFT- GGA. To minimize possible interactions between a defect and its periodic images, we carefully evaluated the formation energies by changing the supercell size, i.e., 192+n, 400+n, 480+n, 560+n, and 672+n atom supercells, where n is the number of interstitials, for I 1 I 2, I 3 I 6, I 7 I 10, I 11 I 13, and I 14 I 16 clusters This work, while 400+n, 480+n, 560+n, and 640+n atom supercells, respectively, for I 3 I 4, I 5 I 6, I 7 I 8, and I 9 I 10 Chainlike. respectively. The predicted values of 3.80, 2.79, 2.06, and 1.85 ev, respectively, for the I, I 2, I 3, and I 4 clusters are close to those from previous DFT-GGA calculations. 28,33 The predicted formation energies exhibit an oscillating trend as the cluster size varies, with strong minima at n=4 and 8, consistent with inverse modeling of experiments. 42,43 When n 8, as can seen in Fig. 3, the compact clusters are more stable than or at least comparable to the chainlike, elongated clusters where a dumbbell interstitial is added to the previously relaxed cluster in the series, see Ref. 30. Our study also implies that the compact configurations would be hard to be transformed to the well-ordered chainlike configurations by fully destroying the stable fourfold I 4 structure. That is, our TBMD simulation at 1100 K shows that the I 4 + I structure remains nearly unchanged for over 20 ps, whereas the I 3 compact cluster easily collapses when it captures an additional interstitial, which would further rearrange to the stable I 4 structure. This suggests that the stable I 4 and I 8 compact clusters would kinetically and/or thermodynamically hamper the formation of small chainlike clusters n 10. When n 10, the elongated geometry becomes more stable than the compact shape, although it commonly consists of strained 110 edges which make it less favorable for smaller clusters. Our results are consistent with earlier experiments which suggested the occurrence of the compact to elongated transition at n ,43 Based on the calculation results, we discuss the nucleation and growth of large, extended defects. One possible mechanism may involve evolution from the fourfold compact to chainlike elongated structures at n 10, followed by the capture of additional interstitials at 110 edges. Another growth mechanism may involve the elongated fourfold I 11 and I 12 clusters identified in this work. When n 10, given the wellordered structure with high thermal stability we hardly ex

4 SANGHEON LEE AND GYEONG S. HWANG FIG. 4. Color online Minimum energy structures for I 12 I 16 clusters, which illustrates a preferred growth along the 110 direction when n 10. The strain energy distribution around the I 12 cluster is also shown at three different levels high 0.25 ev =black blue, medium ev =grey gold, low 0.15 =white, based on strain energy values from KT potential calculations, together with the isosurfaces of occupied orbitals near Fermi level E F, as indicated shaded region in the total density of state TDOS plot in the upper right panel. Here, the Fermi level is positioned at the top of the collection of electron energy levels. pect interconversion between the elongated fourfoldcoordinated clusters this work and the chainlike ones. Figure 4 upper left panel shows the local strain field around the fourfold I 12 cluster and the isosurfaces of occupied orbitals near the Fermi level as indicated in the DOS plot, upper right panel. The larger strain and the isosurface near the 110 edges indicate that they are more active than the 311 and 233 edges. Our TBMD simulations indeed demonstrate that an additional interstitial placed around the I 12 cluster is preferentially captured at either 110 edge to form the fourfold I 13 cluster. Figure 4 also shows the minimum-energy configurations for I 14, I 15, and I 16 clusters which have fourfold coordination, grown from the I 12 cluster. The I 13, I 14, I 15, and I 16 formation energies per interstitial are predicted to be 1.63, 1.63, 1.60, and 1.50 ev, respectively. Note that the I 16 structure with C 2h symmetry is the extended form of I 12 with an additional core unit in the 110 direction. By capturing additional interstitials, the interstitial defect further grows preferentially along the 110 direction. The result suggests that I 12 can serve as an effective nucleation center for the growth of larger extended defects. C. Growth to {311} extended defects We finally look at the possible link of the small fourfold and chainlike clusters to extended defects with a 311 habit FIG. 5. Color online Three different defect core structures which are infinitely long in the 110 direction: a I 12 -like as shown for the fourfold I 12 cluster identified in this work ; b chainlike; c 311. Grey gold balls indicate more distorted atoms than the rest of the lattice atoms in white. plane. In Fig. 5, we compare the I 12 -like a, which represents the core structure of the fourfold I 12 and I 16 clusters identified in this work and chainlike b core structures with a probable 311 core structure c. The 311 core consists of four six-membered rings in the middle layer and six five- and six seven-membered rings in the outer layers, as proposed by earlier studies Other 311 core configurations have also been predicted, 30,49 52 but somewhat similar to the structure considered here. For understanding the link between small clusters and 311 defects, hence it might be unnecessary to consider other possible 311 core configurations. For each repeating unit, the I 12 -like and 311 cores consist of four interstitials, while the chainlike one has two. Compared to the I 12 -like core, the chainlike and 311 cores are twice longer in the 110 and 233 direction, respectively. The I 12 -like and 311 cores exhibit a trace of 311 habit plane when they are repeatedly placed along the 233 direction, but the chainlike one does not. In addition, transformation from the I 12 -like to 311 core can be expected via structural relaxation in the 233 direction. However, it appears rather unlikely that the stable chainlike core would reconfigure to the 311 core, particularly considering its twice longer length in the 110 direction. That is, the chainlike 311 reconfiguration requires that the chainlike structure should shrinks by half in the 110 direction while stretching to twice in the 233 direction, which is expected to hardly occur. The less dense 311 core is found to be more relaxed than the I 12 -like core, as evidenced by its lower formation energy. The predicted formation energies of the 311, I 12 -like and chainlike cores are 1.11, 1.29, and 1.28 ev, respectively, provided the cores are infinitely long in the 110 direction, as shown in Fig

5 GROWTH AND SHAPE TRANSITION OF SMALL SILICON FIG. 6. Color online Minimum energy configurations for the 110 edges of a I 12 -like, b chainlike, and c 311 -core containing clusters which are elongated along the 110 direction. Black blue balls indicate highly distorted atoms in the 110 edges, and gray gold balls represent more distorted atoms than the rest of the lattice atoms in white. For the sake of comparison, we also examine the relative stability of I 12 and I 16 clusters with the 311 core state, as the 311 core is energetically more stable than the I 12 -like core. As shown in Fig. 6 c, the 110 edges can be fourfold coordinated, while each edge initially contains two dangling bonds. The fourfold coordination lowers the I 12 formation energy by 1.1 ev, over the initial structure with dangling bonds. Using CRN-MMC simulations we also confirm that the fourfold edges can favorably be formed for larger clusters with the 311 core structure. However, we find that the edge structure is more strained than that with the I 12 -like core Fig. 6 a. For the I 12 and I 16 clusters, our calculations predict the 311 -core containing structures with C i symmetry to be 1.4 and 0.8 ev, less favorable than the I 12 -like core containing ones. Figure 7 shows the relative stability of larger clusters with the I 12 -like, chainlike, and 311 cores which we approximate using E f n = E end f + n n / E c f /n, where E end f is the 110 edge energy, E c f is the core energy per interstitial for the infinitely elongated structure, and n / indicate the number of edge atoms. For the I 12 -like and 311 cases, taking the core formation energies of E c f =1.29 and 1.11 ev, the corresponding edge energies are estimated to be E end f =8.74 and ev, respectively, as fitted for I 12 and I 16. Note that both cases have four edge atoms, i.e., n / =4. Similarly, for the chainlike FIG. 7. Predicted formation energies per interstitial of I 12 -like, chainlike, and 311 core containing interstitial clusters as a function of size n using E f n = E f end + n n / E f c /n, where E f end is the 110 edge energy, E f c is the core energy per interstitial for the infinitely elongated structure, and n / is the number of edge atoms. The I 12 -like represents the core structure of the fourfold I 12 and I 16 clusters identified in this work, the chainlike indicates the core structure that consists of dumbbell interstitials as also described in Fig. 3, and the 311 represents the 311 core structure in which all atoms are fourfold coordinated see the text. Here, the edge energies were obtained by fitting to DFT-GGA values for the formation energies of smaller clusters, i.e., I 12 and I 16 for the I 12 -like and 311 cases while I 8, I 9 and, I 10 for the chainlike case. The predicted values are based on E f c =1.29, 1.28, and 1.11 ev and E f end =8.74, 6.83, and ev for the I 12 -like, chainlike and 311 cases, respectively, and n / =4 for the I 12 -like and 311 cases and n / =2 for the chainlike case. case we obtain E f c =1.28 ev and E f end =6.83 ev, where n / =2 the chainlike structure has two edge atoms. According to the result, the 311 core structure becomes favored when the cluster size is greater than 20 atoms, below which the I 12 -like core containing structure appears prevailing. This suggests a possible structural relaxation from the I 12 -like to the 311 core state as the interstitial defect grows larger than n 20 along the 110 direction. We also construct an extended defect with a 311 habit plane by repeatedly placing the 110 elongated structure with the 311 core along the 233 direction. The predicted formation energy per interstitial is E f =0.89 ev, in good agreement with ev as reported for extended 311 defects by previous studies. 53 A further investigation is underway to understand atomistic mechanisms regarding how extended 311 defects grow in the 110 and 233 directions. IV. SUMMARY We have determined stable compact geometries for small interstitial clusters n 10, demonstrating that stable I 4 and I 8 compact clusters would kinetically or/and thermodynamically inhibit the formation of chainlike, elongated clusters. When the cluster size is greater than 10 atoms, our calculations show that elongated structures become more favorable energetically, although they commonly consist of highly strained 110 edges which make them less favorable for

6 SANGHEON LEE AND GYEONG S. HWANG smaller clusters. In particular, the newly discovered fourfoldcoordinated I 12 state is found to serve as an effective nucleation center for large extended defects. For the first time, this theoretical work provides explicit support for earlier experiments which suggested the occurrence of the compact to elongated transition at n 10. In addition, the predicted formation energies per interstitial exhibit an oscillating trend with strong minima when n=4 and 8, while decreasing with cluster size in general, consistent with earlier inverse model studies based on experiments. While the fourfold I 12 -like core is energetically less favorable than a typical 311 defect core, our theoretical study suggests the possible occurrence of further structural relaxation from the I 12 -like to the 311 core state as the interstitial defect grows larger than n 20 along the 110 direction. ACKNOWLEDGMENTS We acknowledge Semiconductor Research Corporation, National Science Foundation CAREER-CTS , and Robert A. Welch Foundation F-1535 for their financial support. We would also like to thank the Texas Advanced Computing Center for use of their computing resources. *Author to whom correspondence should be addressed: gshwang@che.utexas.edu 1 G. D. Watkins, Phys. Rev. B 12, E. J. H. Collart, K. Weemers, N. E. B. Cowern, J. Politiek, P. H. L. Bancken, J. G. M. van Berkum, and D. J. Gravesteijn, Nucl. Instrum. Methods Phys. Res. B 139, S. Takeda, Jpn. J. Appl. Phys., Part 2 30, L A. Agarwal, T. E. Haynes, D. J. Eaglesham, H.-J. Gossmann, D. C. Jacobson, J. M. Poate, and Y. E. Erokhin, Appl. Phys. Lett. 70, A. Claverie, B. Colombeau, B. de Mauduit, C. Bonafos, X. Hebras, C. Ben Assayag, and F. Cristiano, Appl. Phys. A: Mater. Sci. Process. 76, J. L. Benton, S. Libertino, P. Kringhøj, D. J. Eaglesham, J. M. Poate, and S. Coffa, J. Appl. Phys. 82, D. C. Schmidt, B. G. Svensson, M. Seibt, C. Jagadish, and G. Davies, J. Appl. Phys. 88, S. Libertino, S. Coffa, and J. L. Benton, Phys. Rev. B 63, M. Nakamura and S. Murakami, J. Appl. Phys. 94, P. K. Giri, Semicond. Sci. Technol. 20, D. Pierreux and A. Stesmans, Phys. Rev. B 71, D. J. Eaglesham, P. A. Stolk, H.-J. Gossmann, and J. M. Poate, Appl. Phys. Lett. 65, L. H. Zhang, K. S. Jones, P. H. Chi, and D. S. Simons, Appl. Phys. Lett. 67, G. S. Hwang and W. A. Goddard, Phys. Rev. Lett. 89, G. S. Hwang and W. A. Goddard, Appl. Phys. Lett. 83, G. S. Hwang and W. A. Goddard, Appl. Phys. Lett. 83, C. L. Kuo, W. Luo, and P. Clancy, Mol. Simul. 29, S. Solmi, M. Ferri, M. Bersani, D. Giubertoni, and V. Soncini, J. Appl. Phys. 94, M. Y. L. Jung, R. Gunawan, R. D. Braatz, and E. G. Seebauer, J. Electrochem. Soc. 151, G M. Y. L. Jung, C. T. M. Kwok, R. D. Braatz, and E. G. Seebauer, J. Appl. Phys. 97, S. A. Harrison, T. F. Edgar, and G. S. Hwang, Appl. Phys. Lett. 87, S. A. Harrison, T. F. Edgar, and G. S. Hwang, Phys. Rev. B 72, S. A. Harrison, T. F. Edgar, and G. S. Hwang, Phys. Rev. B 74, S. A. Harrison, T. F. Edgar, and G. S. Hwang, Electrochem. Solid-State Lett. 9, G J. Zhu, T. Diaz dela Rubia, L. H. Yang, C. Mailhiot, and G. H. Gilmer, Phys. Rev. B 54, P. B. Rasband, P. Clancy, and M. O. Thompson, J. Appl. Phys. 79, N. Arai, S. Takeda, and M. Kohyama, Phys. Rev. Lett. 78, J. Kim, F. Kirchhoff, W. G. Aulbur, J. W. Wilkins, F. S. Khan, and G. Kresse, Phys. Rev. Lett. 83, B. J. Coomer, J. P. Goss, R. Jones, S. Öberg, and P. R. Briddon, Physica B Amsterdam , J. Kim, F. Kirchhoff, J. W. Wilkins, and F. S. Khan, Phys. Rev. Lett. 84, S. K. Estreicher, M. Gharaibeh, P. A. Fedders, and P. Ordejón, Phys. Rev. Lett. 86, M. P. Chichkine and M. M. De Souza, Phys. Rev. B 66, D. A. Richie, J. Kim, S. A. Barr, K. R. A. Hazzard, R. Hennig, and J. W. Wilkins, Phys. Rev. Lett. 92, G. M. Lopez and V. Fiorentini, Phys. Rev. B 69, L. A. Marqués, L. Pelaz, P. Castrillo, and J. Barbolla, Phys. Rev. B 71, M. Cogoni, B. P. Uberuaga, A. F. Voter, and L. Colombo, Phys. Rev. B 71, R M. Posselt, F. Gao, and D. Zwicker, Phys. Rev. B 71, A. Carvalho, R. Jones, J. Coutinho, and P. R. Briddon, Phys. Rev. B 72, S. A. Centoni, B. Sadigh, G. H. Gilmer, T. J. Lenosky, T. Diaz dela Rubia, and C. B. Musgrave, Phys. Rev. B 72, Y. A. Du, R. G. Hennig, T. J. Lenosky, and J. W. Wilkins, Eur. Phys. J. B 57, S. Lee and G. S. Hwang, Phys. Rev. B 77, N. E. B. Cowern, G. Mannino, P. A. Stolk, F. Roozeboom, H. G. A. Huizing, J. G. M. van Berkum, F. Cristiano, A. Claverie, and M. Jaraíz, Phys. Rev. Lett. 82, C. J. Ortiz, P. Pichler, T. Fühner, F. Cristiano, B. Colombeau, N. E. B. Cowern, and A. Claverie, J. Appl. Phys. 96, J. P. Perdew and Y. Wang, Phys. Rev. B 45,

7 GROWTH AND SHAPE TRANSITION OF SMALL SILICON 45 G. Kresse and J. Furthmuller, VASP the Guide Vienna University of Technology, Vienna, D. Vanderbilt, Phys. Rev. B 41, T. J. Lenosky, J. D. Kress, I. Kwon, A. F. Voter, B. Edwards, D. F. Richards, S. Yang, and J. B. Adams, Phys. Rev. B 55, Y. Tu, J. Tersoff, G. Grinstein, and D. Vanderbilt, Phys. Rev. Lett. 81, M. Kohyama and S. Takeda, Phys. Rev. B 46, J. Kim, J. W. Wilkins, F. S. Khan, and A. Canning, Phys. Rev. B 55, P. Alippi and L. Colombo, Phys. Rev. B 62, J. P. Goss, T. A. G. Eberlein, R. Jones, N. Pinho, A. T. Blumenau, T. Frauenheim, P. R. Briddon, and S. Öberg, J. Phys.: Condens. Matter 14, P. A. Stolk, H.-J. Gossmann, D. J. Eaglesham, D. C. Jacobson, C. S. Rafferty, G. H. Gilmer, M. Jaraíz, J. M. Poate, H. S. Luftman, and T. E. Haynes, J. Appl. Phys. 81,